In a large bowl, stir together the mashed potatoes, egg, flour, 2 tablespoons of the chives, grated cheese, salt, and pepper until well combined.
Heat a large skillet over medium high heat and add 1 tablespoon of the oil and 1 tablespoon of the butter. When the butter is melted and hot, start making the pancakes. Form ¼ cup of the mashed potato mixture into a ½ inch thick patty. Add them to the hot pan as you form them, fitting in as many as you can without them touching. Cook them for 3 to 4 minutes per side until they are crispy and browned.
Transfer the pancakes as they are cooked to the paper towels. Keep cooking the pancakes until the mashed potato mixture is gone, adding more olive oil and butter as needed. Let each batch of pancakes drain and sit for 2 minutes before serving them hot or warm, with a bit of sour cream and a sprinkle of chives.
Notes
You can make these pancakes ahead of time and store them, uncooked, in a single layer on a baking sheet in the fridge, covered with plastic wrap for up to 2 days. Or, if you want to fry them up ahead of time, you can store them in the fridge in a container, separating the layers with parchment paper or wax paper, for up to 2 days. Reheat them on a baking sheet for about 10 minutes until heated throughout and crispy on the outside in a preheated 350-degree oven.